(In)Existence of Equilibria for 2-Players, 2-Values Games with Concave Valuations

We consider 2-players, 2-values minimization games where the players' costs take on two values, $a,b$, $a<b$. The players play mixed strategies and their costs are evaluated by unimodal valuations. This broad class of valuations includes all concave, one-parameter functions $\mathsf{F}: [0,1]\rightarrow \mathbb{R}$ with a unique maximum point. Our main result is an impossibility result stating that: If the maximum is obtained in $(0,1)$ and $\mathsf{F}\left(\frac{1}{2}\right)\ne b$, then there exists a 2-players, 2-values game without $\mathsf{F}$-equilibrium. The counterexample game used for the impossibility result belongs to a new class of very sparse 2-players, 2-values bimatrix games which we call normal games. In an attempt to investigate the remaining case $\mathsf{F}\left(\frac{1}{2}\right) = b$, we show that: - Every normal, $n$-strategies game has an ${\mathsf{F}}$-equilibrium when ${\mathsf{F}}\left( \frac{1}{2} \right) = b$. We present a linear time algorithm for computing such an equilibrium. - For 2-players, 2-values games with 3 strategies we have that if $\mathsf{F}\left(\frac{1}{2}\right) \le b$, then every 2-players, 2-values, 3-strategies game has an $\mathsf{F}$-equilibrium; if $\mathsf{F}\left(\frac{1}{2}\right) > b$, then there exists a normal 2-players, 2-values, 3-strategies game without $\mathsf{F}$-equilibrium. To the best of our knowledge, this work is the first to provide an (almost complete) answer on whether there is, for a given concave function $\mathsf{F}$, a counterexample game without $\mathsf{F}$-equilibrium.